Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Write an equation in \(x\) and \(y\) that results in the desired translation. Do not use a calculator. The squaring function, shifted 2 units downward and 3 units to the right

Step-by-Step Solution

Answer

The equation is \( y = (x - 3)^2 - 2 \).

Step 1: Identify the Base Function

The base function given is a squaring function, which is represented as \( y = x^2 \).

Step 2: Understand Translation Movements

We need to shift the function 2 units downward and 3 units to the right. A downward shift affects the vertical position of the graph, while shifting to the right affects the horizontal position.

Step 3: Translate Function Downwards

To shift the function 2 units downward, we decrease the \( y \) values by 2. This changes the function to \( y = x^2 - 2 \).

Step 4: Translate Function to the Right

To shift the function 3 units to the right, we recognize that moving to the right requires replacing \( x \) with \( x - 3 \) in our function. Thus, the equation becomes \( y = (x - 3)^2 - 2 \).

Key Concepts

Squaring Function

The squaring function is one of the simplest polynomial functions. It is represented by the formula \( y = x^2 \). This function creates a U-shaped graph known as a parabola. The vertex of this parabola is at the origin (0, 0) when no other transformations have been applied. The line of symmetry for a basic squaring function graph is the y-axis. This standard form of the squaring function is the starting point for various graph transformations, including translations, reflections, and dilations.

Coordinate Shifts

Coordinate shifts refer to the movement of a graph along the x or y-axis. This is key in graph transformations, where we adjust the position of a graph to meet specific conditions. A shift involves adding or subtracting numbers from the coordinates of the graph.

• Horizontal Shifts: Affect the x-values of a graph.
• Vertical Shifts: Affect the y-values of a graph.

In coordinate shifts, the shape of the graph remains the same, but its position changes. This enables us to model situations where the function needs to be aligned differently.

Graph Transformations

Graph transformations are a set of operations that modify the position, shape, or orientation of a graph. They include translations, reflections, dilations, and rotations. Specifically, translations involve shifting the graph without altering its size or shape.

• Translation: Moving the entire graph horizontally, vertically, or both.
• Reflection: Flipping the graph over a specific line, such as the x-axis or y-axis.
• Dilation: Expanding or contracting the graph, which changes its scale.

These transformations help in accurately modeling different scenarios or visualizing mathematical functions in a modified context.

Horizontal Translation

Horizontal translation involves shifting a graph left or right. This type of shift affects the x-values of each point on the graph. To translate a graph horizontally, you replace \( x \) with \( x-h \), where \( h \) indicates the number of units the graph is moved.

• To the right: Use \( x-h \) with \( h > 0 \).
• To the left: Use \( x+h \) with \( h > 0 \).

For our function, shifting 3 units to the right results in \( y = (x - 3)^2 \). This changes only the x-coordinate while maintaining the form of the function.

Vertical Translation

Vertical translation is when a graph is moved up or down along the y-axis. This movement alters the y-values of the graph but keeps the x-values unchanged. To apply a vertical translation, you either add or subtract a constant from the entire function.

• Upward shift: Add a positive number, \( y = x^2 + k \).
• Downward shift: Subtract a positive number, \( y = x^2 - k \).

In the given exercise, moving the squaring function 2 units downward results in \( y = x^2 - 2 \). When combined with horizontal translation, the function becomes \( y = (x - 3)^2 - 2 \), reflecting both translations.